UPS has trucks up and running Thursday -- and hopes that holiday cheer may assuage the frustrations of angry customers.

The shipping giant's backlog left some Americans with gifts missing from under the Christmas tree Wednesday.

"We're terribly sorry," spokeswoman Natalie Black told CNN.

FedEx was late with some deliveries as well, but those were "isolated incidents," the company said in a statement Thursday.

FedEx "experienced no major service disruptions in the week before Christmas despite heavy volume. Every single package is important to us, and we will continue to work directly with customers to address any isolated incidents," the statement said.

On Wednesday, FedEx spokesman Scott Fiedler told The Associated Press, "We're sorry that there could be delays and we're contacting affected customers who have shipments available for pickup."

The U.S. Postal Service had no immediate comment on whether it experienced any similar delays. Spokeswoman Zy Richardson said in an e-mail, "We take great pride in delivering packages for our customers in time for their holiday. ... Our carriers continued to deliver mail on Christmas Day, in many locations, to make sure everyone enjoyed their holiday."

UPS did some package sorting Wednesday, but the company stuck to its plan to make no deliveries on Christmas, giving drivers a break after doing a lot of special deliveries Sunday.

Both UPS and FedEx noted they were handling hundreds of millions of shipments in a compressed holiday shopping time frame, because Thanksgiving was later than usual this year.

FedEx is projected to handle 275 million shipments, and "operated with very high service levels -- over 99% at FedEx Ground, for example -- during our busiest time of the year," Fiedler's statement said.

Black, at UPS, said that last week, during the peak of shipping season, "we projected to deliver more than 132 million packages."

The company got more air shipments than it expected and was already dealing with a backlog from previous severe weather in the Dallas area, Black said.

"Thanks for totally screwing us AND lying about it all day," customer Mo Husseini posted on Twitter in response to a UPS tweet.

Some packages declared 'delivered' weren't, customers say

Some UPS customers said they received tracking notices stating their packages had been delivered. But they weren't.

UPS customer support tweeted back, providing an e-mail address to help get the situation resolved.

But the company had no immediate comment Thursday about what may have happened.

"We know how hard it is for everyone to receive their holiday packages, and we're working around the clock to resolve this issue," Black said Wednesday.

On the company's Facebook page, Melissa Gilbreath wrote that she ordered her sons' gifts on December 9, but the packages have sat in Fort Worth, Texas, since December 18. "I realize it is not about the gifts but that is part of Christmas," she wrote.

But the Facebook page is also filled with messages from people who say they are UPS employees and proud of their work and the company. "We all work really hard and I think people have too high of expectations," Misty Fulton wrote.

Amazon responds

Online retailers who promised customers delivery in time for Christmas have been affected by UPS' problems.

Some customers received messages from Amazon.com alerting them of the "failure in the UPS transportation network." In the message, the company offered to refund shipping charges and provide gift cards.

"Amazon fulfillment centers processed and tendered customer orders to delivery carriers on time for holiday delivery," spokeswoman Mary Osako told CNN. "We are reviewing the performance of the delivery carriers."

"We have a relationship with Amazon. We will definitely work with all of our retailers," Black said. "We're going to make good on our service commitments. What that exactly entails I don't know, but we'll work with them to maintain that relationship."

After January 2, the day the industry processes returns of holiday gifts, the company will examine what happened and gauge its contingency planning and losses, she said.

Weather has caused delays in previous years. Technical glitches have as well, but Black said it's been "quite some time" since that happened.

Medicine didn't arrive

The backlog isn't just affecting Christmas presents.

The Warren family of Heath, Texas, had to go a few days without special medicine their 14-year-old son needs in order to eat. "It was just sitting" at a UPS outlet in Mesquite, where the family finally picked it up, Patrick Warren told CNN affiliate WFAA-TV on Tuesday.

"People are really upset in there. They have security guards," one woman outside the UPS distribution center in Fort Worth told WFAA.